Number theory Mirsky's early research concerned
number theory. He was particularly interested in the
r-free numbers, a generalization of the
square-free integers consisting of the numbers not divisible by any
rth power. These numbers are a superset of the
prime numbers, and Mirsky proved theorems for them analogous to
Vinogradov's theorem,
Goldbach's conjecture, and the
twin prime conjecture for prime numbers.
Linear algebra In 1947, Mirsky was asked to teach a course in
linear algebra. He soon after wrote a textbook on the subject,
An introduction to linear algebra (Oxford University Press, 1955), as well as writing a number of research papers on the subject. In his research, Mirsky provided necessary and sufficient conditions for the existence of matrices of various types (
real symmetric matrices,
orthogonal matrices,
Hermitian matrices, etc.) with specified diagonal elements and specified
eigenvalues. He obtained a tightening of the
Birkhoff–von Neumann theorem with H. K. Farahat stating that every
doubly stochastic matrix can be obtained as a
convex combination of
permutation matrices. In Mirsky's version of this theorem, he showed that at most n^2-2n+2 permutation matrices are needed to represent every n\times n doubly stochastic matrix, and that some doubly stochastic matrices need this many permutation matrices. In modern
polyhedral combinatorics, this result can be seen as a special case of
Carathéodory's theorem applied to the
Birkhoff polytope. He also worked with
Hazel Perfect on the
spectra of doubly stochastic matrices.
Combinatorics In the mid 1960s, Mirsky's research focus shifted again, to
combinatorics, after using
Hall's marriage theorem in connection with his work on doubly stochastic matrices. In this area, he wrote the textbook
Transversal Theory (Academic Press, 1971), at the same time editing a
festschrift for
Richard Rado. He derived conditions for pairs of set families to have simultaneous transversals, closely related to later work on
network flow problems. He also was one of the first to recognize the importance of
transversal matroids, and he showed that transversal matroids can be represented using linear algebra over
transcendental extensions of the
rational numbers.
Mirsky's theorem, a dual version of
Dilworth's theorem published by Mirsky in 1971, states that in any finite
partially ordered set the size of the longest chain equals the smallest number of
antichains into which the set may be partitioned. Although much easier to prove than Dilworth's theorem, it has many of the same consequences. ==References==